In a hearing aid application, the microphone array is typically placed closely to the ear of the hearing aid user to ensure that the array picks up most realistic sound signals for a natural sound perception. Therefore, the transfer functions dm(k) from a target sound source to individual microphones (m=1, 2, . . . , M) vary over hearing aid users, where k is a frequency index. A look vector d(k) is defined as d(k)=[d1(k), dM(k)]T.
In practical applications, the look vector d(k) is unknown, and it must be estimated. This is typically done in a calibration procedure in a sound studio with a hearing aid mounted on a head-and-torso simulator. Furthermore, the beamformer coefficients are constructed based on an estimate dest(k) of the look vector d(k).
As a result of using the look vector estimate dest(k) rather than d(k), the target-cancelling beamformer does not have a perfect null in the look direction, it has a finite attenuation (e.g. of the order of 10-30 dB). This phenomenon allows the GSC to—unintentionally—attenuate the target source signal while minimizing the GSC output signal e(k,n).